Fixed points of group actions on link collapsible simplicial complexes

Abstract We prove that every action of a group on a link collapsible simplicial complex fixes a simplex. Under additional assumptions, the set of points fixed by the action induced on the geometric realization of the complex is contractible. In particular, it follows that the set of fixed points of the geometric realization of a simplicial endomorphism of a link collapsible clique complex is contractible and that the evasiveness conjecture holds for link-collapsible simplicial complexes.